If \(N\)  is a closed manifold and \((p,v)\in SN\)  then by the Klingenberg's lemma [1]} either there is a second distance minimizing geodesic from \(p\)  to \(\gamma _{p,v}({\tau _{\text{cut}}}(p,v))\)  or these points are conjugate to each other along \(\gamma _{p,v}\) . In particular, the geodesic \(\gamma _{p,v}\)  is not a distance minimizer beyond the interval \([0,{\tau _{\text{cut}}}(p,v)]\) . The following lemma extends this result in our case.
